Optimal. Leaf size=32 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {\coth (x)+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3479, 3480, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {\coth (x)+1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3479
Rule 3480
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\coth (x)}} \, dx &=-\frac {1}{\sqrt {1+\coth (x)}}+\frac {1}{2} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{\sqrt {1+\coth (x)}}+\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\coth (x)}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 51, normalized size = 1.59 \[ \frac {-2+(-1-i) \sqrt {i (\coth (x)+1)} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{2 \sqrt {\coth (x)+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 85, normalized size = 2.66 \[ \frac {{\left (\sqrt {2} \cosh \relax (x) + \sqrt {2} \sinh \relax (x)\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right ) - 4 \, \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}}}{4 \, {\left (\cosh \relax (x) + \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 66, normalized size = 2.06 \[ \frac {\sqrt {2} {\left (\frac {2}{\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}} - \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 27, normalized size = 0.84 \[ \frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{2}-\frac {1}{\sqrt {1+\coth \relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\coth \relax (x) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 26, normalized size = 0.81 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{2}-\frac {1}{\sqrt {\mathrm {coth}\relax (x)+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\coth {\relax (x )} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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